| Version 7 |
Version 6 |
| \textbf{Preliminary Data}\\ |
\textbf{Preliminary Data}\\ |
| \textbf{Definition of a Tree:} |
\textbf{Definition of a Tree:} |
| A {\it tree}, is defined here as the underlying space $ |K| $ of a |
A {\it tree}, is defined here as the underlying space $ |K| $ of a |
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finite $ 1 $-connected $ 1 $-dimensional simplicial complex $ K $ and
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finite $ 1 $-connected $ 1 $-dimensional simplicial complex $ K $ with
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| boundary $ \partial{I}^{2} $ of $ I^{2} $. \\ |
boundary $ \partial{I}^{2} $ of $ I^{2} $. \\ |
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| \begin{definition} |
\begin{definition} |
| a \emph{square} $ u:I^{2} \longrightarrow X $ in a topological space $ X $ is \emph{thin} if there |
a \emph{square} $ u:I^{2} \longrightarrow X $ in a topological space $ X $ is \emph{thin} if there |
| is a factorisation of $ u $, $$ u : I^{2} \stackrel{\Phi_{u}}{\longrightarrow} |
is a factorisation of $ u $, $$ u : I^{2} \stackrel{\Phi_{u}}{\longrightarrow} |
| J_{u} \stackrel{p_{u}}{\longrightarrow} X, $$ where $J_{u}$ is a |
J_{u} \stackrel{p_{u}}{\longrightarrow} X, $$ where $J_{u}$ is a |
| \emph{tree} and $ \Phi_{u} $ is \PMlinkname{piecewise linear (PWL)}{GeometricallyAndorAlgebraicallyThinSquares} on the |
\emph{tree} and $ \Phi_{u} $ is \PMlinkname{piecewise linear (PWL)}{GeometricallyAndorAlgebraicallyThinSquares} on the |
| boundary $ \partial{I}^{2} $ of $ I^{2} $. |
boundary $ \partial{I}^{2} $ of $ I^{2} $. |
| \end{definition} |
\end{definition} |
